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Quantitative jacobian determinant bounds for the conductivity equation in high contrast composite media

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  • We consider the conductivity equation in a bounded domain in $ \mathbb{R}^{d} $ with $ d\geq3 $. In this study, the medium corresponds to a very contrasted two phase homogeneous and isotropic material, consisting of a unit matrix phase, and an inclusion with high conductivity. The geometry of the inclusion phase is so that the resulting Jacobian determinant of the gradients of solutions $ DU $ takes both positive and negatives values. In this work, we construct a class of inclusions $ Q $ and boundary conditions $ \phi $ such that the determinant of the solution of the boundary value problem satisfies this sign-changing constraint. We provide lower bounds for the measure of the sets where the Jacobian determinant is greater than a positive constant (or lower than a negative constant). Different sign changing structures where introduced in [9], where the existence of such media was first established. The quantitative estimates provided here are new.


    Erratum: The name of the second author has been corrected from Haun Chen Yang Ong to Shaun Chen Yang Ong. We apologize for any inconvenience this may cause.

    Mathematics Subject Classification: Primary: 35B30, 35B05; Secondary: 35B27, 35R30.


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  • [1] G. S. Alberti and Y. Capdeboscq, Lectures on Elliptic Methods for Hybrid Inverse Problems, vol. 25 of Cours Spécialisés, Société Mathématique de France, 2018.
    [2] G. Alessandrini and R. Magnanini, Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions, SIAM J. Math. Anal., 25 (1994), 1259-1268.  doi: 10.1137/S0036141093249080.
    [3] G. Alessandrini and V. Nesi, Quantitative estimates on jacobians for hybrid inverse problems, Vestnik YuUrGU. Ser. Mat. Model. Progr., 8 (2015), 25-41.  doi: 10.14529/mmp150302.
    [4] G. Alessandrini and V. Nesi, Univalent $\sigma$-harmonic mappings, Arch. Ration. Mech. Anal., 158 (2001), 155-171.  doi: 10.1007/PL00004242.
    [5] H. Ammari, E. Bonnetier, F. Triki and M. Vogelius, Elliptic estimates in composite media with smooth inclusions: An integral equation approach, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 453–495. doi: 10.24033/asens.2249.
    [6] H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, vol. 1846 of Lecture Notes in Mathematics, Springer, 2004. doi: 10.1007/b98245.
    [7] E. S. BaoY. Y. Li and B. Yin, Gradient estimates for the perfect conductivity problem, Archive for Rational Mechanics and Analysis, 193 (2009), 195-226.  doi: 10.1007/s00205-008-0159-8.
    [8] L. Berlyand and H. Owhadi, Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast, Archive for Rational Mechanics and Analysis, 198 (2010), 677-721.  doi: 10.1007/s00205-010-0302-1.
    [9] M. Briane and G. W. Milton, Change of sign of the correctors determinant for homogenization in three-dimensional conductivity, Archive for Rational Mechanics and Analysis, 173 (2004), 133-150.  doi: 10.1007/s00205-004-0315-8.
    [10] Y. Capdeboscq, On a counter-example to quantitative jacobian bounds, Journal de l'École polytechnique - Mathématiques, 2 (2015), 171–178. doi: 10.5802/jep.21.
    [11] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, 2nd edition, Springer Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.
    [12] S. Friedland, Variation of tensor powers and spectra, Linear and Multilinear Algebra, 12 (1982/83), 81-98.  doi: 10.1080/03081088208817475.
    [13] P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.
    [14] H. Kang, K. Kim, H. Lee, J. Shin and S. Yu, Spectral properties of the Neumann-Poincaré operator and uniformity of estimates for the conductivity equation with complex coefficients, J. Lond. Math. Soc. (2), 93 (2016), 519–545. doi: 10.1112/jlms/jdw003.
    [15] R. S. Laugesen, Injectivity can fail for higher-dimensional harmonic extensions, Complex Variables Theory Appl., 28 (1996), 357-369.  doi: 10.1080/17476939608814865.
    [16] Y. Y. Li and M. S. Vogelius, Gradient estimates for solutions of divergence form elliptic equations with discontinuous coefficients, Arch. Rational Mech. Anal, 153 (2000), 91-151.  doi: 10.1007/s002050000082.
    [17] A. D. Melas, An example of a harmonic map between Euclidean balls, Proc. Amer. Math. Soc., 117 (1993), 857-859.  doi: 10.1090/S0002-9939-1993-1112497-9.
    [18] K.-O. Widman, Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations, Math. Scand., 21 (1967), 17–37 (1968). doi: 10.7146/math.scand.a-10841.
    [19] J. C. Wood, Lewy's theorem fails in higher dimensions, Math. Scand., 69 (1991), 166 (1992). doi: 10.7146/math.scand.a-12375.
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